LECTURE

Third Edition

BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS • A. J. Clark School of Engineering •Department of Civil and Environmental Engineering

11 Chapter 4.6 – 4.7

by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220 – Mechanics of Materials Department of Civil and Environmental Engineering University of Maryland, College Park

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 1 ENES 220 ©Assakkaf

Bending of Composite Beams – In the previous discussion, we have considered only those beams that are fabricated from a single material such as steel. – However, in engineering design there is an increasing trend to use beams fabricated from two or more materials.

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 2 ENES 220 ©Assakkaf

Bending of Composite Beams – These are called composite beams. – They offer the opportunity of using each of the materials employed in their construction advantage. Concrete

Steel

Aluminum

Steel

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 3 ENES 220 ©Assakkaf

Foam Core with Metal Cover Plates – Consider a composite beam made of metal cover plates on the top and bottom with a plastic foam core as shown by the cross sectional area of Figure 26. – The design concept of this composite beam is to use light-low strength foam to support the load-bearing metal plates located at the top and bottom.

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 4 ENES 220 ©Assakkaf

Foam Core with Metal Cover Plates tm

Foam Core

Figure 26

hf Metal Face Plates tm b

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 5 ENES 220 ©Assakkaf

Foam Core with Metal Cover Plates – The strain is continuous across the interface between the foam and the cover plates. The stress in the foam is given by

σ f = Efε ≈ 0

(53)

– The stress in the foam is considered zero because its modulus of elasticity Ef is small compared to the modulus of elasticity of the metal.

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Slide No. 6

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

ENES 220 ©Assakkaf

Composite Beams „

Foam Core with Metal Cover Plates – Assumptions: • Plane sections remain plane before and after loading. • The strain is linearly distributed as shown in Figure 27.

Slide No. 7

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

ENES 220 ©Assakkaf

Composite Beams „

Foam Core with Metal Cover Plates y M Compressive Strain Neutral Axis

x

Tensile Strain Figure 27

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 8 ENES 220 ©Assakkaf

Foam Core with Metal Cover Plates – Using Hooke’s law, the stress in the metal of the cover plates can be expressed as y σ m = εEm = − Em (53) ρ but Em / ρ = M / I x , therefore

σm = −

My Ix

(54)

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 9 ENES 220 ©Assakkaf

Foam Core with Metal Cover Plates – The relation for the stress is the same as that established earlier; however, the foam does not contribute to the load carrying capacity of the beam because its modulus of elasticity is negligible. – For this reason, the foam is not considered when determining the moment of inertia Ix.

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Slide No. 10

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

ENES 220 ©Assakkaf

Composite Beams „

Foam Core with Metal Cover Plates – Under these assumptions, the moment of inertia about the neutral axis is given by I NA

  h f t m  2  bt   = m (h f + t m )2 ≅ 2 Ad = 2 bt m  2   2   2

(55)

– Combining Eqs 54 and 55, the maximum stress in the metal is computed as σ max =

M (h f + 2t m )

bt m (h f + t m )

2

(56)

Slide No. 11

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

ENES 220 ©Assakkaf

Composite Beams „

Foam Core with Metal Cover Plates – The maximum stress in the metal plates of the beam is given by tm

Foam Core hf

Metal Face Plates

σ max =

M (h f + 2t m )

bt m (h f + t m )

2

(56) tm b

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 12 ENES 220 ©Assakkaf

Example 1 A simply-supported, foam core, metal cover plate composite beam is subjected to a uniformly distributed load of magnitude q. Aluminum cover plates 0.063 in. thick, 10 in. wide and 10 ft long are adhesively bonded to a polystyrene foam core. The foam is 10 in. wide, 6 in. high, and 10 ft long. If the yield strength of the aluminum cover plates is 32 ksi, determine q.

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 13 ENES 220 ©Assakkaf

Example 1 (cont’d) The maximum moment for a simply supported beam is given by qL2 q(10 ×12 ) = = = 1800q 8 8 2

M max

When the composite beam yields, the stresses in the cover plates are

σ max = Fy = 32,000 psi

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 14 ENES 220 ©Assakkaf

Example 1 (cont’d) Substituting above values for Mmax and σmax into Eq. 56, we get σ max =

M (h f + 2t m )

bt m (h f + t m )

32,000 =

Or

2

1800q (6 + 2 × 0.063) 2 10(0.063)[6 + 0.063]

q = 67.2

lb lb = 806 in ft

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 15 ENES 220 ©Assakkaf

Bending of Members Made of Several Materials – The derivation given for foam core with metal plating was based on the assumption that the modulus of elasticity Ef of the foam is so negligible,that is, it does not contribute to the load-carrying capacity of the composite beam.

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 16 ENES 220 ©Assakkaf

Bending of Members Made of Several Materials – When the moduli of elasticity of various materials that make up the beam structure are not negligible and they should be accounted for, then procedure for calculating the normal stresses and shearing stresses on the section will follow different approach, the transformed section of the member.

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 17 ENES 220 ©Assakkaf

Transformed Section – Consider a bar consisting of two portions of different materials bonded together as shown in Fig. 28. This composite bar will deform as described earlier. – Thus the normal strain εx still varies linearly with the distance y from the neutral axis of the section (see Fig 28b), and the following formula holds: y εx = − (57) ρ

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 18 ENES 220 ©Assakkaf

Transformed Section y

y M

1

εx = −

y

σ1 = −

ρ

ρ

εx

N.A 2

Figure 28 (a)

E1 y

σx σ2 = −

(b)

E2 y

ρ

(c)

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 19 ENES 220 ©Assakkaf

Transformed Section – Because we have different materials, we cannot simply assume that the neutral axis passes through the centroid of the composite section. – In fact one of the goal of this discussion will be to determine the location of this axis.

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Slide No. 20

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

ENES 220 ©Assakkaf

Composite Beams „

Transformed Section We can write: σ 1 = E1ε x = −

E1 y

(58a)

σ 2 = E 2ε x = −

E2 y

(58b)

ρ

ρ

From Eq. 58, it follows that dF1 = σ 1dA = −

E1 y

dF2 = σ 2 dA = −

E2 y

ρ

ρ

dA dA

(59a) (59b)

LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 21 ENES 220 ©Assakkaf

Transformed Section – But, denoting by n the ratio E2/E1 of the two moduli of elasticity, dF2 can expressed as dF2 = −

(nE1 )y dA = − E1 y (ndA) ρ

ρ

(60)

– Comparing Eqs. 59a and 60, it is noted that the same force dF2 would be exerted on an element of area n dA of the first material.

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LECTURE 11. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4.6 – 4.7)

Composite Beams „

Slide No. 22 ENES 220 ©Assakkaf

Transformed Section – This mean that the resistance to bending of the bar would remain the same if both portions were made of the first material, providing that the width of each element of the lower portion were multiplied by the factor n. – The widening (if n>1) and narrowing (n